It will provide protection from Lightning strikes that originate 40 units from placement spot. That's a 40 unit radius and 80 unit diameter horizontally/vertically, and a 28.28 unit radius and 56.56 unit diameter diagonally. Lightning may originate outside the lightning rod protection radius, and strike target inside the protection zone.
To connect protection areas with multiple Rods using simple math, players should leave 14 Pitchfork tiles between each Rod horizontally/vertically. Imagine each zone as a square of 56 units in length by width instead of a circle for easier comprehension. If each Rod is placed 14 Pitchfork tiles between one another horizontally/vertically, it will create a protection grid with minimum excess overlaps.
(See the Console/Commands page for instructions on measuring distance; distance can be measured with a Pitchfork's 4x4 dig area).
The Lightning Rod also provides a small radius of light (but not heat) after absorbing a Lightning bolt, which is just bright enough to prevent Charlie from attacking. The light lasts for a few Days and will go dark as soon as morning comes. You may see some benefit to placing your lightning rod close to your fire pit so that even in otherwise total darkness and without another light source you can still add fuel for a fire.
In all DLCs, the cost to craft a Lightning Rod is increased to 4 Gold Nuggets.
If a Lightning Rod and a Volt Goat are in close vicinity in the Reign of Giants DLC, they both have an equal chance of being struck by Lightning.